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系統識別號 U0002-0307201317280800
中文論文名稱 Reeb定理上的Morse理論及其推廣
英文論文名稱 Morse Theory to Reeb's Theorem and its Generalization
校院名稱 淡江大學
系所名稱(中) 數學學系碩士班
系所名稱(英) Department of Mathematics
學年度 101
學期 2
出版年 102
研究生中文姓名 李容瑄
研究生英文姓名 Jung-Hsuan Lee
學號 600190069
學位類別 碩士
語文別 英文
口試日期 2013-06-26
論文頁數 27頁
口試委員 指導教授-余成義
委員-謝忠村
委員-林吉田
中文關鍵字 Morse引理  Morse定理  Reeb定理 
英文關鍵字 Morse lemma  Morse Theorem  Reeb's Therem 
學科別分類 學科別自然科學數學
中文摘要 本文先介紹可微流型之定理及各種基本性質,也初略介紹具Riemann測度之可微流型。
本文主要內容在使用Morse引理及Morse定理,重新述明Reeb定理之證明。這裡我們使用Surgery引理,處理邊界問題。
英文摘要 In this thesis, we want to use Morse Theorem to prove Reeb's Theorem. Before showing the proof of these theorems, we need to review some basic properties of a differentiable manifold M with a differentiable structure. In general, if we define some functions from M (or its subspace) to real value, the difference between manifold and coordinate space should be considered. Every point we choose must send to a coordinate subspace first. So defining a coordinate system is helpful to deal with any functions on manifold M.
The main result we review is to prove Reeb's Theorem using Morse Lemma and Morse Theorem. Here we use a surgery lemma to prove disjoint union of two spaces, matched along their common boundary.
We also show how to construct a homotopy equivalence between manifold M and a n-sphere, for all dimension n is larger or equal to 1.
論文目次 Contents
0 Introduction 1
1 Di erentiable manifold 3
2 Non-degenerate Critical Point And Hessian 7
3 Smooth Vector Field on Manifold 9
4 Homotopy 13
5 Morse Lemma 15
6 Main Theorem 18
7 State of Reeb's Theorem(construct a homeomorphism between M and S^n) 23
8 Conclusion and Remarks 26
參考文獻 [1] William M. Boothby. An Introduction to Differentiable Manifolds and Rieman-
nian Geometry. ACADEMIC PRESS,INC, Orlando,Florida, second edition,
1986.
[2] J.Milnor. On manifolds homeomorphic to the 7-sphere. Annals of Mathematics,
64:399{405, 1956.
[3] James R. Munkres. Elements of Algebraic Topology. rst edition.
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